Embedded newform 69.2.g.a.20.4

• Label: 69.2.g.a.20.4
• Level: $69$
• Weight: $2$
• Character: 69.20
• Analytic conductor: $0.551$
• Analytic rank: $0$
• Dimension: $60$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 69 = 3 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	69.g (of order \(22\), degree \(10\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(0.550967773947\)
Analytic rank:	\(0\)
Dimension:	\(60\)
Relative dimension:	\(6\) over \(\Q(\zeta_{22})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{22}]$

Embedding invariants

Embedding label			20.4
Character	\(\chi\)	\(=\)	69.20
Dual form			69.2.g.a.38.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.666243 - 0.0957913i) q^{2} +(0.369589 + 1.69216i) q^{3} +(-1.48428 + 0.435824i) q^{4} +(1.76154 - 2.03293i) q^{5} +(0.408330 + 1.09199i) q^{6} +(0.360284 + 0.560612i) q^{7} +(-2.17168 + 0.991773i) q^{8} +(-2.72681 + 1.25081i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 60 q - 11 q^{3} - 10 q^{4} - 14 q^{6} - 22 q^{7} - 11 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/69\mathbb{Z}\right)^\times\).

\(n\)	\(28\)	\(47\)
\(\chi(n)\)	\(e\left(\frac{5}{22}\right)\)	\(-1\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).

(See \(a_n\) instead)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.666243	−	0.0957913i	0.471105	−	0.0677347i	0.0973262	−	0.995253i	\(-0.468971\pi\)
							0.373779	+	0.927518i	\(0.378062\pi\)
\(3\)	0.369589	+	1.69216i	0.213382	+	0.976969i
							\(5\)	1.76154	−	2.03293i	0.787785	−	0.909153i	−0.209860	−	0.977731i	\(-0.567301\pi\)
0.997646	+	0.0685787i	\(0.0218464\pi\)
\(7\)	0.360284	+	0.560612i	0.136174	+	0.211891i	0.902643	−	0.430391i	\(-0.141624\pi\)
							−0.766468	+	0.642282i	\(0.777988\pi\)
\(11\)	0.634904	−	4.41586i	0.191431	−	1.33143i	−0.636794	−	0.771034i	\(-0.719740\pi\)
							0.828224	−	0.560397i	\(-0.189351\pi\)
\(13\)	−2.96398	−	1.90483i	−0.822060	−	0.528306i	0.0606857	−	0.998157i	\(-0.480671\pi\)
							−0.882746	+	0.469851i	\(0.844308\pi\)
\(17\)	−3.35165	−	0.984134i	−0.812895	−	0.238687i	−0.151241	−	0.988497i	\(-0.548327\pi\)
							−0.661654	+	0.749809i	\(0.730145\pi\)
\(19\)	1.63223	+	5.55887i	0.374459	+	1.27529i	0.904192	+	0.427127i	\(0.140474\pi\)
							−0.529732	+	0.848165i	\(0.677708\pi\)
\(23\)	3.88197	+	2.81608i	0.809447	+	0.587193i
							\(29\)	−0.554055	+	1.88694i	−0.102885	+	0.350396i	−0.994805	−	0.101798i	\(-0.967540\pi\)
0.891920	+	0.452194i	\(0.149359\pi\)
\(31\)	2.95780	+	6.47667i	0.531236	+	1.16324i	0.965008	+	0.262222i	\(0.0844552\pi\)
							−0.433771	+	0.901023i	\(0.642817\pi\)
\(37\)	−0.669271	+	0.579927i	−0.110027	+	0.0953394i	−0.708137	−	0.706075i	\(-0.750464\pi\)
							0.598110	+	0.801414i	\(0.295919\pi\)
\(41\)	−1.34913	−	1.16903i	−0.210699	−	0.182571i	0.543118	−	0.839657i	\(-0.317244\pi\)
							−0.753816	+	0.657085i	\(0.771789\pi\)
\(43\)	−4.37103	−	1.99618i	−0.666576	−	0.304415i	0.0532384	−	0.998582i	\(-0.483046\pi\)
							−0.719814	+	0.694167i	\(0.755773\pi\)
\(47\)			0.638992i			0.0932066i	0.998913	+	0.0466033i	\(0.0148397\pi\)
							−0.998913	+	0.0466033i	\(0.985160\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	69.2.g.a.20.4	yes	60
3.2	odd	2	inner	69.2.g.a.20.3	✓	60
23.15	odd	22	inner	69.2.g.a.38.3	yes	60
69.38	even	22	inner	69.2.g.a.38.4	yes	60

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
69.2.g.a.20.3	✓	60	3.2	odd	2	inner
69.2.g.a.20.4	yes	60	1.1	even	1	trivial
69.2.g.a.38.3	yes	60	23.15	odd	22	inner
69.2.g.a.38.4	yes	60	69.38	even	22	inner